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Furthermore, the reaction of different benzothiazole derivatives with diphenylphosphine oxide obtained the desired products (10a-10e) in moderate to good yields (65%-87%) (Figures ). We proposed a mechanism for the construction of C-P bond. The excited state K-1* interacted with A and deprotonated to give a radical B and a radical cation C. Then, the radical cation C was oxidized by O2 and regenerated ground state D to complete the photocatalytic cycle. Meanwhile, radical species B attacked benzothiazole E to furnish radical intermediate F, which underwent rearomatization by oxidation and deprotonation to obtain the final product (Scheme 2).
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In summary, novel supramolecular ALHSs based on AIE-active macrocycle K-1 was constructed, which successfully achieved one-step and two-step FRET processes. Thanks to the typical AIE property, undesirable fluorescence quenching is effectively avoided during the FRET process. By contrast with the non-cyclic AIEgen K-2, the ALHSs based on rigid macrocyclic molecule K-1 exhibited higher energy transfer efficiency, which was calculated to be 82.6% for the K-1/PBTB system and 66.4% for the K-1/PBTB/Z1 system, respectively. Bright white light emission can be successfully achieved at the ratio of K-1/PBTB = 200:1 and K-2/PBTB = 500:3 with CIE coordinates of (0.33, 0.34), respectively. More importantly, the ALHS based on macrocycle K-1 exhibited higher photocatalytic activity for the cross-dehydrogenated coupling (CDC) reaction under white-light in water, with yields of up to 87%. Thus, the flexibility of this novel supramolecular strategy makes macrocyclic AIEgen K-1 a promising candidate for the construction of efficient ALHSs for photocatalysis.
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Over the course of the past couple of decades, Density Functional Theory (DFT) has grown into the workhorse of quantum chemistry and materials science, enabling a broad spectrum of applications. An excellent accuracy/cost trade-off can in principle be achieved with the help of DFT, under the condition that a suitable exchange-correlation functional approximation (F xc ) is selected. F xc are usually ranked using Jacob's ladder, where the most complex approximations can be found at the top. As a rule of thumb, functionals higher up the ladder tend to be more accurate, though this is most certainly not a universal rule that holds across application domains. As such, it is common practice in chemistry to determine the most suitable F xc for a given type of application based on a benchmarking against a limited set of either experimental or high-level wave-function computed, data points.
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One of the earliest examples of a benchmarking database was curated by Pople and co-workers, during their efforts to develop new quantum chemical methods for accurate energy calculations. Their database, dubbed "G1", consisted of a mere 31 atomization energies of atoms, ions, and organic molecules. This database later evolved into the G2/97, G3/99, and G3/05 databases; the second incarnation of which was subsequently used to fit the empirical parameters of the popular B3LYP functional. Since then, many more advanced and extensive -compiled -benchmarking datasets have been conceived, to allow the development and assessment of new methods across a large variety of chemical problems, e.g., Database 2015B by Thrular and co-workers,, 12 MGCDB84 database by Head-Gordon and co-workers, and the GMTKN55 database by Grimme and co-workers. Despite the ever-expanding scope of modern benchmarking databases, it is not always clear to which extent the data points present in them are truly representative of the chemical spaces they are intended to describe. As previously noted by others, 14,15 the make-up of these datasets is often strongly biased, e.g., towards highly stable, easily accessible compounds for experimental datasets, and towards small, easily computable molecular systems for computational ones. Consequently, the transferability of exchange-correlation functional accuracy across chemical space is not inherently guaranteed, so a benchmarking study may lead to incorrect conclusions if no attention is paid to this issue. For example, it is not necessarily true that the functional approximation yielding the most accurate results on a specific small benchmarking set will also perform the best for the broader surrounding chemical space. Even worse, while a benchmarking study on a small set of chemical systems for a given type of application may give the impression that a given set of functional approximations is sufficiently accurate -indicating that it should be possible to extract clear chemical trends from data computed at the corresponding levels of theory -this assumption may not hold if (some of) these same functionals struggle with specific regions of the chemical space under consideration that were not part of the original benchmarking set.
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It is important to underscore that other researchers have previously thought about some of the issues outlined above, and several potential (partial) solutions have been proposed over the years. One strategy that has been explored is the use of a recommender system, i.e., a machine-learning (ML) model that will predict which F xc ought to perform best for the specific (molecular) system under consideration. Typically, a metric, indicative of the accuracy of the F xc is designed, and then a multi-task regression model is trained. 17,18,21 Upon inference, the predicted scores of the individual functionals are ranked, and the model will recommend the best one. While arguably the most robust approach to deal with variations in functional performance across chemical space, it is important to appreciate the extreme computational cost of generating reliable training datasets for such recommender models.
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For example, DELFI, a recently developed recommender system for functional selection for excited state calculations of small organic molecules, required the construction of an initial dataset of 828 282 single-point TD-DFT calculations and over 21 000 reference values. An unrelated, and much less expensive, strategy to reduce the odds of suboptimal DFT functional selection consists of 'mindless benchmarking'. This approach, pioneered by Grimme and co-workers, involves random generation of artificial molecular geometries, after which a representative set is sampled with a particular focus on diversity. In this manner, the introduction of biases during the benchmarking dataset construction is limited, increasing the likelihood that the resulting set is at least somewhat representative of the broader chemical space that it aims to describe. Despite the promise of this approach for benchmarking some specific chemistry-related tasks more reliably, the strategy is not ideally suited for others. For example, when benchmarking reaction kinetics, one typically aims to identify a good functional approximation for specific reaction classes, across a well-defined scope of the accessible chemical space, so that a purely randomized approach is hard to implement.
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In this work, a principled, data-driven strategy to assess the transferability of benchmarking results and to construct more challenging and/or representative datasets will be presented, based on active learning. With a particular focus on the simulation of pericyclic reactions, we train here a machine learning model on a small pre-existing benchmarking dataset, to identify regions of chemical space for which activation energies computed with different functional approximations diverge the most. Next, a Bayesian optimization (BO) algorithm is applied to identify the reactions in the chemical space surrounding the initial dataset exhibiting the biggest variations in DFT computed activation energies. The selected reactions then ought to be representative of more challenging patches of chemical space than those present in the training set.
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Iteratively acquiring these reactions and updating the model accordingly, we demonstrate that its generalizability across the defined chemical space improves rapidly, i.e., the discrepancy between predicted and computed variations in activation energy values of the acquired points gradually declines, and convergence is reached after only a handful iterations, cor-responding to the acquisition of fewer than 100 new data points. With the final, validated model at our disposal, new benchmarking datasets with particularly challenging reactions can be curated, and an informed estimate of the maximal errors across chemical space can be inferred.
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Overall, we believe that the presented approach provides a computationally inexpensive, and highly data-efficient, strategy to improve the reliability of -and confidence inbenchmarking efforts. Furthermore, our strategy is easily extensible to other regions of the chemical space, so that the presented work could provide a blueprint for further advances in simulation method selection for future chemical reactivity studies, as well as in the development of new, robust DFT functionals.
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Extracting and curating the initial data from the BH9 dataset BH9 is an extensive and diverse benchmark dataset for reaction and activation energies, composed of 449 chemical reactions belonging to nine types common in organic chemistry and biochemistry. The molecular species in BH9 comprise main-group elements (H, C, N, O, F, P, S, and Cl), plus B and Si. The pericyclic subset, on which we will focus throughout this study, consists of 140 Diels-Alder (DA), [3+2] cycloaddition (DC), electrocyclic, [3,3] rearrangement (RR), [6+4], [4+6], [8+2] and [2+2+2] cycloaddition reactions. We focus on this specific reaction class because of its chemical and biological importance, as well as the good understanding and relative robustness of the corresponding mechanisms.
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Within this work, we aim to identify increasingly challenging reactions, i.e., reactions for which the activation energy exhibits the highest variability across several functionals. Two common quantities can in principle guide us to this end, the range and the standard deviation (σ) of the computed activation energies. We considered the range to be less useful here, as it does not convey any information regarding how dispersed the values truly are, i.e., a high range can be obtained because a single functional is an outlier for a given reaction data point, while this datapoint may still result in a narrow distribution of (activation) energy values overall. σ on the other hand is a direct measure of how dispersed the values are around the mean. For this reason, we selected it as the target quantity throughout this study.
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The first step towards the design of a broader chemical space around the benchmarking dataset consisted of converting the geometries in the pericyclic subset of BH9 into SMILES, using the functionalities of RDKit. 30 Subsequently, reactive cores, i.e., collections of atom pairs undergoing a change in bonding throughout the reaction, were selected. Only those cores with a repeated occurrence, and for which a high σ in the DFT computed activation energies across functionals is obtained (a cutoff of 4.80 kcal/mol was set), were selected.
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Both the mentioned criteria were considered essential. As an illustration, the [8+2] addition exhibits the highest standard deviation in the activation energy values across the BH9 subset (7.52 kcal/mol), but this type of reaction occurs only once (in two different regioisomers), and hence this reaction subclass was rejected. On the other hand, 53 data points of the Diels-Alder [4+2] addition are present in BH9, with the reaction between naphthoquinone and a functionalized 1,3 butadiene being the most divergent with a standard deviation in the activation energy over 6.50 kcal/mol (First core of the Diels-Alder box in Fig. ).
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In total, 10 reactive cores were selected as templates in this manner, respectively divided into five DA, three DC, and two RR ones (the corresponding reaction SMILES can be found in Section S1 of the Supporting Information). An initial set of 9 substituents to decorate the selected reactive cores were identified from the same subset of the BH9-extracted pericyclic reactions. Additionally, 5 extra cores were included to enable the generation of fused rings.
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Two types of fingerprints were investigated as potential reaction representations, the differential reaction fingerprints (DRFP), and the differential Morgan fingerprints. Both belong to the family of circular fingerprints, where the chemical information of the surrounding atoms up to a radius r is encoded as a machine-readable representation. The DRFP algorithm creates a binary fingerprint based on the symmetric difference of two sets, containing the circular molecular n-grams generated from the molecules listed left and right from the reaction arrow, respectively. The Morgan fingerprints of the reaction SMILES were computed by subtracting the molecular fingerprints of the reactants from those of the products.
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A limitation of fingerprint-derived representations is that they are inherently local in nature, i.e., they do not capture long-range changes in the molecular structure. Since it became clear from a preliminary analysis of the BH9 subset that such phenomena, e.g., the formation or disintegration of a macrocycle throughout the reaction, can have an outsized effect on the functional approximation divergence (vide infra), we thus decided to concatenate a final 6-bit vector, encoding the information about the size of new rings formed/broken. Overall, the fingerprints were generated using radius r = [1, 2, 3] and dimensionality dim = [256, 512, 1024, 2048].
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A fully automated reaction profile computation workflow based on autodE and Gaus-sian16 was set up to acquire new reaction data points. As the level of theory for the DFT calculations, CAM-B3LYP in combination with the 6-311++G** basis set in the gas phase was used for the final optimization of all stationary points. The maximum number of conformers generated for single molecules and transition states was set to 1000, using an RMSD cut-off of 0.1 Å to exclude identical conformers. Conformers were ranked based on a loose optimization at CAM-B3LYP/6-31G*, and the lowest energy one was selected for refinement with the larger basis set. The D3 dispersion correction with Becke-Johnson damping was used in all cases. An IRC confirmation at the same level of theory was consistently performed on the final TS. The final functional and basis set were selected with the aim of approaching the level of theory of Prasad et al. as closely as possible.
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BO is a direct application of Bayes Theorem, and consists of two main components: the construction of a surrogate model to approximate the black-box function f to be optimized, and an acquisition function for deciding the next samples to evaluate. Pseudo-code for the BO algorithm used can be found in the ESI; for more details about the foundations and/or applications, we refer to references. Surrogate model
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A surrogate model is a trained ML model used to predict the objective function of the reaction in the generated chemical space. Three types of surrogate models were explored, namely K-Nearest Neighbor, Random Forest, and XGBoost. The first two were implemented in Python using the packages Scikit-learn, and the latter has been implemented with the help of a dedicated package, XGBoost. More details regarding the architectures can be found in Section S5.1 of the ESI.
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The acquisition function is used to select the most promising evaluations to perform next, i.e., which reaction profile to acquire with the help of our automated reaction profile computation workflow. Acquisition functions are typically derived by considering the mean, µ(x), and standard deviation, σ(x), of the surrogate model predictions, either by considering an ensemble of ML models trained on various distinct data splits, or by analyzing the distribution of the individual estimator predictions (e.g., when random forests are selected). In this work, the upper confidence bound (UCB) has been used specifically:
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where β is an explicit hyperparameter to balance the ratio between exploitation and exploration, higher beta values will result in the prioritization of zones for which the model is uncertain, i.e., exploration is favored, while lower values will result in the prioritization of sampling data points with high predicted performance, i.e., exploitation is favored.
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First, we started with an exploratory data analysis of the pericyclic subset of BH9 and the connection between specific structural elements and the observed variation in the activation energy values computed across the set of exchange-correlation functionals probed. Figure illustrates that DA is the most common reaction type in the dataset, followed by electrocyclic reactions, RR, and DC. At first glance, it seemed that reactions involving highly conjugated systems exhibited a more significant activation energy variability across several functionals, while intramolecular reactions appeared to exhibit a less pronounced variability (Fig. ).
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Benchmarking subset 1 consists of all the reactions with σ below 3.5 kcal/mol; subset 2 consists of all reactions with a higher σ. Comparing the performance of the different functionals, with respect to the DLPNO-CCSD(T) computed reference, between the two subsets, we observe remarkable differences, both in qualitative and quantitative terms (Fig. ). While the 3 best-performing functionals, ωB2-PLYP, ωB97M-V, and M06-2X are only negligibly affected by evaluating a different subset, the ordering of the remaining functionals changes profoundly. For example, the functional ranking in fourth place in terms of performance for benchmarking subset 1, ωB97X-V, drops 3 places when evaluated on subset 2. Remarkably, while BLYP and B3LYP are seemingly quite robust when evaluated on subset 1, appearing halfway in the functional ranking and resulting in an acceptable MAE of 2-3 kcal/mol, for subset 2, their mean errors almost triple (to 8-9 kcal/mol), rendering these functionals the second and fourth least reliable functionals of all the ones tested, respectively. Adding empirical dispersion corrections partially remedies the remarkable failure of these functionals, but even then, significant performance losses are still observed (cf. Section S3 in the ESI).
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The observation above is particularly concerning because the latter functionals are still commonly used in reactivity studies. Oftentimes, a handful of benchmarking data points, indicating that these functionals do not perform dramatically worse than more modern functionals, are used as justification for their selection, as the BLYP and B3LYP functionals tend to also be among the fastest to evaluate. This preliminary analysis however already demonstrates that caution is needed in this regard, as trends emerging in local patches of chemical space may not always hold beyond them.
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As described in Section 3, three surrogate model architectures in combination with two different fingerprint representations were explored. We evaluated the performance of each surrogate model/fingerprint combination using 6-fold nested cross-validation on the 140 cycloaddition reactions of the original BH9 dataset. Morgan fingerprints clearly outperform DRFP, a lower radius (r =1 and r =2) results in lower errors than higher ones (r =3), and increasing the number of bits, decreases the error in our calculations.
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was set to favor the exploration of the unknown regions, and a maximum of 15 points were acquired per round. Besides the acquisition function, we also applied a filter based on structural diversity, i.e., we required a minimal cosine distance between the fingerprints of new reactions being selected for acquisition. By applying this diversity filter, we ensured that the various reactive cores were each sampled as part of the campaign. The motivation for this approach stems from the observation that some cores intrinsically exhibit a higher σ than others; e.g. the highest value for the Diels-Alder reactions amounts to 6.68 kcal/mol, while for the [3,3] rearrangements, the highest value in the original BH9 subset amounts to a mere 5.15 kcal/mol. Without diversity-based filtering, we would have sampled almost exclusively reactions for the reaction cores resulting in the most divergent activation energy values in the initial dataset, and only minimal exploration would have been performed for the other cores at best. The settings of each acquisition round can be found in Section S2.1
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The performance of our model after each acquisition round was monitored with the help of nested 6-fold cross-validation. We observed that the (intrapolation) MAE consistently oscillated around the initial value of 0.52 kcal/mol (R 2 =0.78), and did not improve throughout the campaign (cf. Figure ). This however does not mean our model was not improving overall/becoming more robust: the accuracy of the predictions across the wider constructed chemical space rose rapidly, indicating improvements in the generalizability power across the chemical space of the model. From 1.15 kcal/mol in round one, the MAE on σ of the activation energy values gradually and monotonously decreased to 0.60 kcal/mol by round seven, i.e., the mean error was cut by over half its initial value. Starting from round five, a smooth leveling-off in the error reduction could be observed, and the prediction error in the final round approached the error obtained during (intrapolative) cross-validation. Both of these observations suggest that convergence had essentially been reached by this point.
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It should be noted here that the fact that a model, trained on a combination of the original BH9 subset and the acquired reactions, generalizes much better across the designed chemical space, suggests that this resulting, expanded dataset is significantly more representative of the selected chemical space than a (subset of) the original benchmarking dataset. As already indicated, after 7 rounds, the exploration stage of the BO campaign was stopped, as the σ of the newly acquired data points had barely improved with respect to the previous rounds, and the MAE of the predictions appeared to approach convergence. Some examples of the best reactions emerging from the exploration stage are shown in Figure , with the selected substituents highlighted. Substituents that extend the delocalization, such as phenyl or ester groups, as well as voluminous groups introducing steric hindrance, turn out to be key for increasing the σ of the activation energy values.
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Upon conclusion of the exploration stage, a final exploitation round, consisting of the acquisition of 30 points (with a β value of 1.0), was initiated. From this batch, the reaction profiles were successfully acquired for 26 reactions. The MAE of the predictions on this round was 0.61 kcal/mol, in line with the error obtained in the last exploration round, confirming that model convergence had indeed been reached. From the final pool of 108 acquired reactions, 70 were selected to form our new benchmarking dataset. A table containing the list of reactions for this new dataset organized by type, and the reference reaction energy and barrier heights can be found in Section S2.4 of the ESI. DLPNO-CCSD(T) reference values were computed for the acquired reactions (see Section S2.4 of the ESI for the methodology used), and the deviation for every functional was computed. The resulting accuracies are presented graphically in Figure , together with the accuracies for the original BH9 cycloaddition dataset (values are presented in Table of ESI).
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Overall, the individual errors, as well as the relative ordering of the functionals have changed dramatically. Three functionals that appeared in the middle of the pack when evaluated on the original BH9 subset, perform best on our newly acquired data points: ωB2-PLYP18, PBE-QIDH, and RSX-QIDH. Remarkably, these three functionals buck the trend of all the other functionals, by actually performing better on the new data compared to the original data.
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For most of the other functionals, the deterioration in performance is far from uniform, and at times spectacular. For example, M06, PBE0, and B2-PLYP, which appeared fairly robust when subdividing the Diels-Alder reactions of BH9 into two benchmarking subsets (cf. Figure ), exhibit a mean error at least three times larger when evaluated on our active learning dataset instead of on BH9. In absolute terms, the deterioration of BLYP, CAM-B3LYP and BHandHLYP is even worse, but as indicated before, adding dispersion corrections remedies the situation to some extent (cf. Section S3 of the ESI). The ωB97M-V and M06-2X functionals on the other hand remain relatively robust, though the MAEs still deteriorate by approximately 2 kcal/mol. Because of this deterioration, the former functional, which performed best on the original BH9 subset, now only comes in fourth place when ranking the functionals based on performance.
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Validation of exchange-correlation functionals is an essential task in (computational) chemistry, and the design of benchmark datasets is key to this end. In this work, a new approach, based on Bayesian optimization and active learning, for generating more diverse, unbiased, benchmarking datasets is presented. Starting from an initial model trained on 140 data points, our strategy enables the identification of challenging regions of chemical space within only a handful of iterations. Sampling new data points in these regions reveals that we have successfully pushed the mean errors for most functionals higher, and a completely different picture of the performance is obtained, with the ranking of some functionals being changed dramatically. It is important to note that the strategy followed in this paper is easily extensible to other relevant chemical properties (as well as functionals), and is extremely data-efficient.
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While we consider this work in the first place as a proof of principle, instead of a comprehensive benchmarking study in its own right, we can nevertheless interpret the obtained results and provide some guidelines about which functionals to select when studying cycloaddition reactions. ωB2-PLYP18, and PBE-QIDH turn out to be extremely robust functionals, exhibiting MAEs on our set of acquired reactions lower than 2.0 kcal/mol. Alternatives (and cost-effective) options could be the hybrid ωB97M-V, and the meta-GGA B97M-V with an MAE of 2.6 and 3.5 kcal/mol, respectively.
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The main conclusion of this study, however, is that current benchmarking datasets such as BH9 are not necessarily representative of their surrounding chemical spaces. This is an issue that has received limited attention up to this point but needs to be addressed to improve the reliability of -and the confidence in -DFT studies.
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Tautomers are structural isomers that interconvert from one form to another. The most common, and best-known, type of tautomerism occurs by proton migration. Other forms that involve equilibrium between ring and chain form with and without proton migration are called ring-chain and valence tautomers. In textbooks and literature, the term "readily" is often used to describe the process. However, this is not correct in all cases. In the case of solid forms of drug molecules, the transformation from one form to another is very slow due to the difficult migration of protons.
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To speed up and make less expensive the typically very expensive and time-consuming drug development process, numerous computational tools have been developed. Preclinical studies involve estimating absorption, distribution, metabolism, and excretion (ADME) properties and optimizing new chemical entities. Issues with these pharmacokinetic properties, as well as toxicity, have been part of the reasons for failure of drug candidates. All these properties affect the pharmacodynamic action of a drug molecule. A good candidate should have balanced ADME and good potency. This mini-review discusses the impact of tautomerism on pharmacokinetics, pharmacodynamics properties, formulation, and racemization. We emphasize here that tautomerism is not a novel phenomenon of drugs in the pill or the body.
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The effect of tautomerism on bioavailability has not been explored much, based on the small number of pertinent research articles. Molecules capable of tautomerism are expected to behave differently when exposed to solvents or gastrointestinal fluids. They may show different tautomeric preferences based on pH, temperature, and microfluids of the intestine.
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Tautomerism in the field of metabolism has likewise received little attention, though most drugs are metabolized in the body. The most commonly represented tautomer is considered for analyzing the metabolism of drug molecules. Different tautomers can be metabolized by enzymes or converted non-enzymatically at different rates. Warfarin, an anti-coagulant agent, exists in two enantiomeric forms and has at least 40 tautomers ranging from open chain to ring tautomers. Studies on warfarin and analogue compounds have suggested that tautomeric S-warfarin forms are metabolized in ring form. The pharmacodynamics effect will differ if minor or major tautomers are metabolized rapidly. After one or two rounds of initial oxidation, drugs may go for tautomerization due to the addition of a hydroxyl group or epoxide in the structure. For example, cyclophosphamide is converted to 4-OH-cyclophosphamide via hydroxylation, then this ring tautomer is converted to a chain tautomer (Figure ). Another interesting example is daclatasvir, which is a selective inhibitor of the hepatitis C virus. After δ-oxidation at the pyrrolidine moiety, daclatasvir is prone to tautomeric equilibrium with a chain tautomer (aminoaldehyde) via opening of the pyrrolidine ring. This is converted to a stable metabolite by attacking the aldehyde group with imidazole nitrogen. Mechanism-based inhibition (MBI) is a special type of inhibition where a metabolite formed from a drug is quite reactive and can bind to a metabolizing enzyme covalently or non-covalently, leading to temporary or permanent loss of metabolizing enzyme activity, hence affecting the metabolism of other drugs (drug-drug interaction). These end metabolites can have the potential for tautomerization, and only one of the tautomers leads to MBI (lapatinib nitroso tautomer, (Figure )).
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Human serum albumin (HSA) is one of the abundant transporter proteins present in the blood. HSA has multiple binding sites due to its ability to adapt to accommodate drug molecules. There is a general lack of studies as to which types of drugs or specific tautomers binds in particular HSA sites. However, there are many examples where one enantiomer is reported to bind tightly and the other weakly or with a different binding strength. Drugs usually form reversible complexes with HSA by electrostatic and hydrophobic forces, and strong binding may significantly affect the pharmacokinetic properties.
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Warfarin primarily exists as the open-chain 4-hydroxycoumarin tautomer in aqueous environments. According to spectroscopic studies, the ionized form of the open-chain tautomer binds with HSA to distribute the drug to the body. The binding study of 3,4,7,8-Tetrahydroxyflavone with HSA indicated that this molecule exists in both keto and enol tautomers; however, in HSA bound complex, both tautomers are deprotonated by releasing proton from hydroxyl group. One of the tautomers has a significant interaction in one of the sites, and another tautomer has a weak interaction in the other site. Overall, the interaction of drugs with human HSA is a complex process and plays a significant role in drug transport, distribution, drug-drug interactions, clearance, and toxicity.
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Most drugs lose their pharmacological activities mainly through metabolic transformation, which increases their polarity. This results in metabolites with high water solubility that are readily renally excreted. Other excretion routes are feces, sweat, and bile. We found no direct information on the excretion of tautomeric metabolites or preference of one tautomer over other in excretion.
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The pharmacokinetics study of the tautomer of cefotetan revealed that it is converted to a tautomer with antibacterial activity similar to the parent drug, predominantly in the urine of rabbits and monkeys. However, this tautomer was not observed in the plasma of studied animals including humans (Figure ). Tautomerism may play a role in converting metabolites to final or intermediate metabolites that facilitate their easy excretion. For example, in the case of the drug elzasonan, tautomerism was reported via the formation of the indole iminium ion metabolite through ring closure and rearrangement via proton migration between two carbons, and this metabolite was not detected in urine but excreted through feces. 3. Pharmacodynamics Tautomerism can affect the binding of a drug to its target and the subsequent drug action, including the drug's efficacy and potency. Erythromycin is a frequently prescribed antibiotic that exists in three tautomeric forms (one ketone and two cyclic hemiketals) (Figure ). It has been proven that the ketonic form is active and recognized by ribosomes, inhibiting protein synthesis. The inactive tautomer can be present in up to 20% in the gastrointestinal tract.
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Therefore, erythromycin has to be given in a correspondingly larger dose. The keto-enol tautomers of curcumin-based molecules were potent at both BACE-1 and GSK-3β targets, while diketo tautomer-based molecules were less active or inactive. In some cases, tautomers exist in equilibrium with cationic, anionic, and zwitterionic forms, and it is difficult to determine which form may be responsible for drug action. For example, edaravone is present in 50% anionic form and the rest as a mixture of three neutral tautomeric forms (Figure ). Avobenzone is the only FDA-approved ultraviolet filter molecule that absorbs harmful UVA radiation. The protective effect and stability of this molecule is tautomer-dependent. The keto-enol tautomer protects from UVA, while diketo tautomer is susceptible to photodegradation. 11
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Designing a molecule for targets of CNS diseases presents significant challenges as the blood-brain barrier (BBB) restricts the migration of molecules to the brain more strongly than to other body compartments. If the designed molecule is amenable to tautomerism, it may have multiple forms under physiological conditions. It should be present in sufficient amounts or exclusively to allow entry of the predominant tautomer. Edaravone is known to exist in anionic and tautomeric forms; however, a slight modification to the phenyl ring led exclusively a keto tautomer which had good BBB permeability, while the replacement of the methyl group by trifluoromethyl led to only the enol form with poor BBB permeability. 12
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The formation of benzene-oxide is widespread in drugs containing a phenyl ring. The bicyclic benzene-oxide metabolite can equilibrate with the corresponding seven-membered oxepine by valence tautomerism. Oxepine can progress to ring opening to yield a reactive alpha, betaunsaturated aldehyde, and trans-trans-muconaldehyde, which are known to have myelotoxicity. The hepatotoxicity of methylenedioxyphenyl containing (Figure ) molecules (safrole, and myristicin) is associated with quinone and its methide tautomer, which are generated during phase I metabolism. CYP1A2 activates safrole into the o-quinone and its methide tautomer, which leads to the inactivation of the enzyme by tightly binding with the enzyme's active site. Photodegradation products of the diketo tautomer of avobenzone have phototoxic effects and cause photoirradiation, restricting its application in sunscreen products. 11
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There are cases where one polymorphic form has one tautomer while the other has a different tautomer or mixture. Using mechano-chemical treatment, a thermodynamically less stale tautomeric form can be transformed into a more stable tautomer. The processed solid state can be studied using FTIR, powder XRD, HPLC, UV, and optical microscopy. In addition to these, broadband dielectric spectroscopy is a very sophisticated method used to detect and monitor tautomers of drugs. Different dosage forms may also affect the ratio of tautomeric forms, e.g., the thickness of the nanometric layer. The diketo form of curcumin is favored in low concentrations in liposomes. In higher concentrations, the keto-enol form is dominant in liposome-based nanoformulation. Guest systems like cyclodextrin and beta-cucurbits for pharmaceuticals have shown huge potential for improving their stability, solubility, compatibility with other ingredients, and dissolution rate. If a drug is tautomeric, these carriers selectively can entrap on tautomer over others with improved physiochemical properties. Moreover, the degree of derivatization of the carrier system can show selectivity for one tautomer over another. 17
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Rosiglitazone's antidiabetic action is related to the (S)-enantiomer. A carbonyl group is present in the proximity of the chiral centre; therefore, keto-enol tautomerism can convert the (S)-enantiomer to the (R)-enantiomer or vice versa. This rate of interconversion is very high after S-oxidation of rosiglitazone. One of the notorious examples is thalidomide, which shows therapeutic activity by the S-enantiomer, while the R-enantiomer shows teratogenicity. These enantiomers undergo racemization at physiological conditions very easily via keto-enol tautomerism.
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Studies of thermodynamic properties (stability, solubility) are crucial in understanding and explaining drug pharmacokinetic properties and behavior during preclinical studies using in vitro and in vivo models. Tautomers are characterized using traditional spectroscopy and crystallography (X-ray) methods. Broad band spectroscopy has been instrumental in studying the behavior of tautomers in solid crystal, amorphous, and glass phases of drugs. Most drugs are represented commonly using structure determined by the X-ray method, which may be incorrect as there are many cases where the tautomer of the drug in the solid is different from the tautomer in the aqueous phase or other solvents. This tautomer seen in aqueous solution may furthermore be different under different physiological conditions.
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Understanding the excited state properties of molecules helps describe how they interact with light. These photochemical interactions can include fundamental processes such as photosynthesis, human vision, 2 or photostability . Photochemistry is also important in designing new molecules with certain properties, for example spectral converters for photovoltaics, which are of particular interest in this study. Using an interplay between their excited states, certain molecules can up-or down-convert wavelengths of light to improve photovoltaic efficiency. Unfortunately, existing spectral conversion molecules have low efficiency, so further exploration is required. It is difficult to explore the excited state space of molecules with experimental methods alone, so researchers often turn to computational methods for more detailed study. Traditionally, time-dependent density functional theory (TD-DFT) has been the workhorse for excited state energy calculations. However, TD-DFT can be computationally intensive, and for applications in high-throughput screening or generative design, a faster method must be used.
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TD-DFT often relies on DFT for ground-state calculations of charge density and structure optimization. Recently, work has been done in tight binding as an approximation to DFT to a) Also at Department of Mechanical Engineering, Massachusetts Institute of Technology, 33 Massachusetts Ave, Cambridge, MA, USA improve its computation time while retaining most of its accuracy. Specifically, density functional tight binding (DFTB) was developed in the late 1990's and exhibited a combination of the accuracy of DFT and the efficiency of semiempirical quantum chemistry methods. More recently, the eXtended Tight Binding (xTB) methods were developed to solve the issues with DFTB of extensive parameterization and low transferability. They differ from DFTB methods in that they utilize top-down parameterization, with semiempirical parameters fit to a large dataset rather than computed with first-principles calculations. The primary approximations are considering molecular orbitals to be a linear combination of atomic orbitals (LCAOs), using the local density approximation (LDA) for exchange-correlation energy, and using a truncated Taylor expansion to map density to total energy. To accelerate excited-state calculations, Grimme introduced the simplified Tamm-Dancoff density functional approach (sTDA) as an approximation to TD-DFT. The key approximations of sTDA include simplifications to twoelectron integrals and setting an upper limit to the excitation space, which improve computation time by 2 orders of magnitude. As sTDA was developed to calculate excitation spectra, there is no excited state relaxation component, and only vertical excitation energies can be calculated. The differences between vertical excitation, vertical emission, and adiabatic energy are shown in Figure . In this work, only vertical excitations are considered. The additional computational expense of excited state relaxation is prohibitively slow for to excited state (S 1 ), with various vibrational levels (0-3) depicted for both states. (b) Demonstration of Frank-Condon principle of 0→1 vertical excitation (blue arrow) followed by nuclear re-configuration and 1←0 vertical emission (red arrow). Also shown is the 0→0 transition energy in yellow. (c) The expected experimental excitation/absorption curve (blue) and emission curve (red), along with the theoretical 0→0 energy difference (dashed yellow line), demonstrating the Stokes shift. Note that in computation, often the energy minimum is used instead of the lowest vibrational level, so the starting energies for excitation and emission may be different in experiment.
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high-throughput workflows. Further, the Stokes shift for rigid molecules should be small, on the order of 0.1 eV. From now on, for concision, the vertical excitation energy will be referred to as the excitation energy or excited state energy. xTB and sTDA can be combined in a workflow called xTB-sTDA, allowing ultrafast computation of excited states. This has been used extensively, with several studies using the method to screen large databases of materials such as copolymers, conjugated polymers, small aromatic molecules, photocatalysts, and organic dyes. However, due to the approximations presented above, there is a tradeoff between accuracy and computational speed. In Grimme and Bannwarth's original paper introducing xTB-sTDA, they reported a mean absolute error (MAE) between xTB-sTDA and coupled-cluster/TD-DFT calculated excited state energies of 0.34 to 0.48 eV, depending on the complexity of the input structure. Even though xTB-sTDA is often used as a first-pass in high-throughput screening, with higherquality computational methods used to evaluate properties of a screened subset of molecules, having an accurate first-pass method is essential to ensure all suitable candidates are included in the suggested subset.
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Therefore, a method of improving the accuracy of xTB-sTDA in a way that preserves its high-throughput characteristics is desired. One approach is to calibrate the results of xTB-sTDA against a higher-accuracy computational methods using machine learning (ML). This type of calibration from a baseline method to a reference method is known as the delta-ML or ∆-ML approach , and has been applied widely in the literature for various computational techniques. For example, it has been used for calibrating ground state energies and structures from semiempirical methods to DFT and coupled cluster (CC) accuracy, and calibrating DFT molecular dynamics simulations or potential energy surfaces to CC accuracy. Recently, the ∆-ML approach has been applied to calibrate excited state properties, for example calibrating photoemission spectra from DFT to G 0 W 0 , and calibrating excited state energies with TD-DFT against CC methods and against experiment. For xTB-sTDA, a few studies have used a linear calibration technique to correct excited state energies, instead of ML. However, the improvement from linear calibration was low, with a mean absolute error (MAE) of around 0.2 eV, compared to around 0.1 eV for the ML studies presented above. To the authors' knowledge, no previous study has applied ML to calibrate xTB-sTDA. Due to the promising potential of ML to increase the accuracy of baseline methods, this work presents a ML calibration of the excited state energy levels output by xTB-sTDA, with the motivation of more efficient exploration of excited state space. As mentioned previously, existing spectral conversion materials utilize excited states to up-or down-convert photon energy, but have low efficiency due to (a) energy level misalignment which leads to energy loss and (b) low absorptionto-emission probability. This study focuses on the first issue, making it easier to accurately predict energy levels and therefore design high-efficiency spectral conversion materials. The following sections present the methods and results of our excited state energy calibrations.
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xTB was originally parameterized from the spincomponent-scaled coupled cluster (SCS-CC2) method and TD-DFT, so one of these would be a natural choice as the reference computational technique. Coupled-cluster (CC) methods are typically the most accurate, predicting excitation energies within 0.1 eV of experiment. However, because of the computational expense of CC, it is difficult to generate a large amount of data using these methods. Instead, TD-DFT is generally the workhorse for excited state calculation, despite its relatively high errors, with an MAE of 0.2 to 0.4 eV (depending on the functional) compared to experiment or the theoretical best estimate. A general-purpose exchange-correlation functional is B3LYP , which has been shown to have an MAE around 0.25 eV, while LDA, GGA, and other hybrid functionals have higher error. While B3LYP performs well for local-ized densities, range-separated hybrid functionals such as CAM-B3LYP are increasingly used for delocalized densities such as those in excitations as they include a long-range correction. However, these range-separated hybrid functionals are more computationally expensive than B3LYP. Due to its accuracy and efficiency, B3LYP is the most commonly used functional in computational molecular chemistry. In this work, we choose TD-DFT based on B3LYP as the reference method, for several reasons. First, we require large chemical diversity in our training set, and most of the existing molecular excited state databases use TD-DFT. The largest databases, namely PubChemQC and QM-symex, use B3LYP. Second, B3LYP was used in previous works using linearly calibrated xTB-sTDA and is used extensively in machine learning and high-throughput screening studies. Third, while B3LYP is less accurate than range-separated hybrid functionals, it is not significantly worse. Since xTB-sTDA is semi-empirical, it is often used as a first-pass screening and naturally has inconsistencies and false positive/negative errors. Calibrating to B3LYP accuracy should lower the rate of these errors. For these reasons, the reference method was chosen to be TD-DFT with B3LYP.
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Specifically, the training sets for the ML models considered in this study were derived from the existing PubChemQC (PCQC) and QM-symex databases. For concision, we will use the (functional/basis set) notation to describe the level of theory used in calculations. PCQC includes the first 10 singlet excited state energies (S 1-10 ) for 3.5M molecules computed using B3LYP/6-31G(d) for ground state optimization and B3LYP/6-31+G(d) for excitation. Similarly, QM-symex computes both S 1-10 and the first 10 triplet excited state energies (T 1-10 ) for 173k molecules using B3LYP/6-31G(2df,p) for ground state optimization and B3LYP/6-31G for excitation.
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We are interested in calibrating both singlet (S 1 ) and triplet (T 1 ) excited state energies output by xTB-sTDA, but PCQC does not include triplet excitations. Calculating triplet excitations for 3.5M molecules independently would be prohibitively expensive, so it was necessary to determine which molecules in PCQC would be relevant to spectral conversion applications and therefore have interesting excited state properties. To extract such molecules from PCQC, a literature scraping workflow was developed. We used the SCO-PUS API 48 to obtain abstracts of articles tagged with "triplettriplet annihilation" or "singlet fission" keywords. Then, we used ChemDataExtractor to extract molecule names from the abstracts. We then used the PubChem API 50 to convert molecule names into PubChem CIDs and conduct a 2D Tanimoto-coefficient based similarity search among Pub-Chem molecules to expand the molecular space of interest. We then cross-referenced the identified molecules against PCQC to get the singlet energies, and triplet energies were independently generated with TD-DFT using equivalent set-tings to PCQC. Overall, this process allowed us to select 10k molecules of interest from PCQC, named SCOP-PCQC (after SCOPus-PCQC). To balance the 10k molecules in SCOP-PCQC, a 10k subset of QM-symex was randomly selected and named QM-symex-10k.
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However, this 10k molecule subset of PCQC may be too small for ML model training. In addition, a model trained on only molecules relevant to spectral conversion may have poor out-of-domain performance. Therefore, we turn to active learning to add diverse molecules to the training set with further sampling of PCQC. Here, we use active learning techniques to evaluate regions of chemical space where the ML model is uncertain. Active learning is an ML technique often used to sample unexplored regions of state space. Our implementation uses a trained ensemble of ML models to measure uncertainty of the remaining chemical space. Specifically: first, a 10-ensemble ML model was trained on the 10k SCOP-PCQC molecules to directly predict S 1 and T 1 energies. Then, the ensemble was used to predict S 1 and T 1 energies on the remaining 3.5M molecules in PCQC. The 100k molecules with the highest uncertainty (variance in ensemble prediction) were chosen as an expansion to SCOP-PCQC, labeled SCOP-AL-Exp, for each of S 1 and T 1 . This process helps ensure broad applicability of the ML model. More details about the active learning process are available in SI Section V.
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To evaluate the generated models, various test datasets were used. First, we used 10-fold cross-validation with 80%/10%/10% training/validation/test splits to quantify indomain accuracy. In k-fold cross-validation, k non-overlapping test sets are generated, and models are trained on the remaining 90% of data. Validation sets are also non-overlapping and are used to prevent overfitting.
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To prove broader applicability, external test sets were also compiled. 1143 molecules from Wilbraham et al.'s paper on Mapping the optoelectronic property space of small aromatic molecules (MOPSSAM) was used, 143 from their calibration training set and 1000 randomly selected from the remaining 250k molecules. 10k molecules from Fallon et al.'s paper on indolonaphthyridine thiophene (INDT) derivatives were also used as they are promising candidates for singlet fission. 1000 molecules from Abreha et al.'s VERDE database (Verd-eDB) were used, as the classes of molecules identified (porphyrins, quinones, dibenzoperylenes) are relevant for various green chemistry excited state applications. Finally, to truly test the broader applicability of the model, another active learning cycle was run on PCQC. Using a training set composed of the 10k molecules from SCOP-PCQC plus the 200k molecules from SCOP-AL-Exp, an ensemble ML model was generated and used to evaluate uncertainty on the remaining PCQC molecules. 100k of the highest uncertainty molecules for each of S 1 and T 1 were chosen as the last test set, labeled PCQC-AL. Additional information about active learning is in SI Section V). To visualize the training and test datasets, we plotted the locations of the datasets in chemical space. We used uniform manifold approximation and projection (UMAP), a dimensionality reduction technique that reduces the high-dimensional space of chemical structure into 2 dimensions for ease of visualization. We use the Jaccard-Tanimoto similarity between Morgan fingerprints of molecules as a measure of proximity in chemical space. We first generated a global UMAP based on all molecules, then categorized them into (a) training and (b) test, and colored them based on their dataset, shown in Figure .
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-A few trends become apparent from this visualization. As seen in Figure (a), SCOP-PCQC is primarily localized to two regions, while the active learning expansions have broader coverage of chemical space. Many molecules in the SCOP-AL-Exp set are localized around the SCOP-PCQC molecules, suggesting that despite the chemical similarity of structures, their excited state energies may be significantly different. We further observe that the QM-symex-10k dataset provides a uniform sampling of the overall QM-symex dataset, and that the QM-symex datasets are significantly different from PCQC. Turning to test datasets, we note that the INDT dataset is significantly different from both the PCQC and QM-symex based training datasets. VerdeDB has some molecules outside but near the PCQC training sets, while others are within the PCQC space. Finally, both MOPSSAM datasets seem to lie within the PCQC training set space, implying good predictive performance is expected.
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Since the supervised ML model takes in molecular structure and excited state data, we must obtain excited state data for all molecules. xTB-sTDA data was all independently generated. A 3D structure was first intialized using OpenBabel's gen3d function for a short conformer search and preliminary geometry optimization. Full ground-state optimization was conducted with GFN2-xTB with the tight threshold and a benzene generalized-Born surface-area (GBSA) solvation model to mimic a non-polar environment. xtb4stda 55 was then used to prepare the wavefunctions output by xTB for sTDA. Finally, sTDA was used to calculate excited-state properties, using an energy threshold of 10 eV. The -t flag was used for triplet excited state calculations. For TD-DFT data, database values were used where available. PCQC had S 1 TD-DFT data, but T 1 data was independently generated. MOPSSAM had S 1 TD-DFT data for the 143 calibration set, but not for the 1000 sampled molecules, so this was independently generated (see Figure for a comparison of MOPSSAM 143 S 1 data vs. S 1 data generated with our workflow, showing virtually identical results). T 1 TD-DFT data was also independently generated. Both INDT and VerdeDB had S 1 and T 1 TD-DFT data available. However, VerdeDB used the M06 functional for calculations, so these molecules were re-calculated with B3LYP.
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Note that while the xTB-sTDA portion of the workflow was standardized, the TD-DFT data was database-dependent. For consistency, only databases that used the B3LYP functional were included in this study, but initial coordinate generation technique and basis sets for (TD)DFT varied for each dataset. The specific settings for each database are shown in Table . Once excited state values using both xTB-sTDA and TD-DFT were either compiled or calculated, they could be fed to the ML model.
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The type and architecture of the ML model must be optimized for performance. The 3 ML models considered were DeepChem's graph convolutional network (DC GCN), DeepChem's message passing neural network (DC MPNN), and Chemprop's directed message passing neural network (CP MPNN). These 3 models were chosen as they are commonly used graph neural networks, which have emerged as a natural choice for molecules where nodes represent atoms and edges represent bonds. The models each use different architectures and methodologies for featurization and property prediction. DeepChem's GCN is based on Duvenaud et al.'s paper which introduced a method to generalize conventional circular fingerprints using convolutional neural networks to generate neural graph fingerprints. DeepChem's MPNN is based on Gilmer et al.'s work which expands upon Duvenaud et al.'s GCN and is better able to identify correlations between node and edge states. Chemprop's MPNN is based on Yang et al.'s work which adds directionality to the message passing step, preventing noisy graph representations. We are interested in comparing each model's performance in our application. The default, out-of-the-box settings for each ML model were used, as described in SI Section III. Calibration of the 1000 molecules in the VerdeDB database was used to compare the different ML models. The small size and relatively homogeneous nature of this dataset makes it suitable for quickly comparing different ML models. Only the SMILES (simplified molecular-input line-entry system ) representation of the molecule was provided as input, and the goal of each model was to accurately predict the S 1 and T 1 error between xTB-sTDA and TD-DFT. Instead of predicting both S 1 and T 1 error simultaneously, two separate single-task models were generated, both using 10-fold cross-validation. For each fold, the trained ML model was used to predict error values of the test set. Then, each molecule's predicted error was added to the xTB-sTDA output to give a calibrated energy, called the xTB-ML value. The xTB-ML values were compared to the TD-DFT reference results by calculating an R 2 score.
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Figure (a) shows the results of comparison for S 1 and T 1 energies. As seen, all ML models vastly outperform the linear calibration method. Between the ML models, CP MPNN performs the best for both T 1 and S 1 . Note that the large variability in R 2 can be explained by the presence of outliers in the test set -since the test set was only composed of 100 molecules (10% of 1k), a few outliers can vastly impact performance. calibrated xTB-sTDA data for (a) S 1 and (b) T 1 energies, with test data from all 10 folds compiled and with outliers removed (full plots with outliers available in SI Section IV). From this analysis, it is evident that CP MPNN performs well in calibrating xTB-sTDA results, even with its default settings.
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To see if the performance could be boosted further, various architectural improvements were attempted. These included increasing the number of epochs (number of iterations to optimize the neural network weights) to 100, conducting hyperparameter optimization (finding the best ML architecture i.e. hidden size, depth, dropout, and number of feed-forward layers), and conducting multi-task training (using a single model to predict both S 1 and T 1 energies simultaneously). More details about these optimization approaches are available in SI Section III.
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The results from these improvements are shown in Figure . As seen, there are only small differences in performance between the default settings and any potential improvements to the ML settings. For T 1 , hyperparameter optimization provides minimal improvement, while including additional features or adding multitasking reduces accuracy. For S 1 , both hyperparameter optimization and multitasking marginally improve performance. There is thus a tradeoff in using multitasking as it could reduce accuracy for T 1 predictions but improve accuracy for S 1 , while also reducing overall computation time. Because of the time savings of the multi-task model and previous works showcasing the benefits of multi-property prediction, this was used for ML for the following sections. Hyperparameter optimization was not performed for the following models, due to the only marginal improvement seen. Based on this analysis, a larger-scale calibration model can now be developed using CP MPNN.
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Combining all of the above methodology, a workflow was developed to create ML models that calibrate the xTB-sTDA energy of molecules against TD-DFT reference. The workflow can be separated into three distinct steps: data generation, model training, and model testing. In the data generation step, S 1 and T 1 excited state energies using TD-DFT and sTDA were either extracted from existing databases or calculated, if necessary. The errors between the energies derived from the two techniques were calculated and used as the groundtruth values that the ML model tried to predict. The SMILES strings were also extracted for molecules and used as a representation of molecular structure.
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In the model training step, ML models were trained to take input data and predict xTB-sTDA vs. TD-DFT error. Two classes of models were trained, one with only the SMILES string as input (Class 1) and another with both SMILES and sTDA energy as input (Class 2). During training, the SMILES string was converted to the graphical representation of the molecule, which was then featurized using an MPNN. If the sTDA energy was used (Class 2), it was concatenated as an extra feature at this step. Then, feature to property prediction was conducted using a feed-forward NN. To improve reliability of results and ensure all molecules were included in the training process, a 10-model ensemble was generated with 10-fold cross-validation using 80%/10%/10% train/validation/test splits. This process resulted in an optimized ensemble ML model for error prediction.
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Once the ML model was trained, it was tested on various datasets. Using the respective inputs, the ML model predicted xTB-sTDA vs. TD-DFT errors for the test molecules. When the errors were added to the original xTB-sTDA values, the final calibrated energies were obtained. These were compared to the TD-DFT-calculated values to get a quantitative measure of accuracy of each ML model.
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Before considering external datasets, ML model performance was evaluated on subsets of the training sets themselves. 10-fold cross-validation (training on 90% of the data and testing on the remaining 10% 10 times with non-overlapping test sets) was conducted separately on the SCOP-PCQC, QM-symex-10k, SCOP-AL-Exp, and QM-symex datasets. Class 1 models (only molecular structure as input) and Class 2 models (both molecular structure and sTDA energy as input) were both tested. Results are compiled in Figure (a), with plots of Class 2 models for SCOP-AL-Exp and QM-symex shown in Figure (b)-(e). As seen, the original data has low accuracy when compared to TD-DFT results, and linear calibration improves the accuracy slightly. However, there is not a clear linear shift due to some groups of molecules located farther from the line of best fit. In contrast, for both datasets, the ML-calibrated values have much lower MAE and demonstrate significant improvements from uncalibrated xTB-sTDA values, especially for T 1 data. The increase in accuracy with ML is likely because ML detects higher-order patterns, allowing groups of molecules to shift locally instead of having to follow a global calibration rule.
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As seen, the ML models performed well in cross-validation. However, it is possible that the ML model only performed well because the datasets were homogeneous, so similar molecules to those in the test set were included in the training set. To evaluate the broad applicability of our model, we used external test sets of molecules not included in either of the datasets above.
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To test broad applicability, a more general ML model is needed. Therefore, an overarching ML model was trained on the 10k SCOP-PCQC molecules combined with the 10k QM-symex-10k molecules, for a total training size of 20k molecules. The overarching ML model was first tested on the MOPSSAM 143 external dataset. As seen in Figure , the ML calibrated xTB-sTDA data matches TD-DFT values better than both the original data and the linearly calibrated data. While the data are sparse, there are a few regions where the improvement is clearly visible. For example, for high S 1 energies, the linear calibration tends to overcorrect, while for low S 1 energies the linear calibration undercorrects. In contrast, the ML model is more flexible and adequately corrects models in both regions. For T 1 energies, the ML model performs similarly to linear calibration with both MAE and R 2 metrics. This is likely because xTB-sTDA nearly always over-predicts the T 1 energy, so calibrating it only requires shifting in one direction, which makes linear calibration sufficient for the task. For S 1 energies, there are both instances of over-and underprediction, which motivates the need for an ML model. However, there is clearly room for improvement in these results, as the ML MAE is still high.
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Two avenues of improvement were pursued -first, using a larger training set, and second, adding additional input data to the ML model. As discussed in the Methods section, a larger training set was generated using active learning to sample regions of chemical space not represented in the 20k training set. The expanded ML model uses 200k molecules chosen with active learning plus all QM-symex molecules (120k), added to the initial 20k training set, for a total of approximately 300k molecules. To distinguish the two ML models generated, the 20k model is named xTB-ML-20k while the expanded 300k model is xTB-ML-300k. The second improvement explicitly included the xTB-sTDA calculated energy as an input to the ML model. As discussed previously, for these models, called Class 2, the xTB-sTDA energy was concatenated to the generated molecular features during the training process.
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As seen, all ML models improve raw xTB-sTDA values, but to different extents depending on the training and test set considered. For the MOPSSAM and PCQC-AL datasets, using larger datasets with more input data generally improves results. This result is intuitive, as more data allows the ML model to learn more about patterns in the datasets. The lowest MAE obtained was 0.08 eV using the Class 2 xTB-ML-300k model on T 1 energies of MOPSSAM 143.
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For the INDT and VerdeDB datasets, the results are less intuitive. In these cases, the Class 1 xTB-ML-20k ML model, i.e. the model trained on a smaller training set with less input data, performs better. There are several reasons this could be happening. The INDT and VerdeDB datasets are composed specifically of molecules relevant to photon conversion or green chemistry applications. Similarly, the xTB-ML-20k dataset is composed primarily of literature scraped molecules intended for spectral conversion applications, so it is more likely to include molecules similar to those in INDT or Verd-eDB. Although the training set size in xTB-ML-300k is larger, the method of expansion through active learning specifically includes molecules significantly different than the 20k training set, so the model may not increase in accuracy for photon conversion molecules, i.e. is not backwards compatible. In terms of why the Class 2 models perform worse than Class 1 models, this could be related to the nature of the training sets used. Both the 20k and 300k training sets have only a few low-excited state energy molecules. Therefore, the calibration in this region may be inaccurate. As seen in Figure , the INDT dataset is primarily composed of low-energy molecules. Therefore, if the energy is localized by providing the sTDA energy, the molecule may undergo an inaccurate calibration. This localization is minimized if the calibration is done solely based on molecular structure, allowing a more accurate calibration.
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Most of the best-performing ML models result in an MAE of less than 0.20 eV. However, the one dataset with large MAEs is the PCQC-AL T 1 energies. As seen in Figure , most of the calculated energies follow a general linear trend. However, there is a large cluster of molecules distinctly separated from the rest that is inflating the MAE. This is expected from the active learning workflow, which selects molecules difficult to predict with the existing model. Naturally, if a subset of these molecules were included in the training set, the overall MAE would likely improve drastically. Regardless, while this test dataset has large errors, by design these are the largest errors one can obtain, and for general PCQC molecules the error should be lower.
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Overall, these results show that the ML models significantly improve raw xTB-sTDA calculated values. In most cases, the best-performing ML model reduces the MAE by more then half. Further, while not shown in Figure , the ML models also consistently outperform linear calibration, showing the benefits of a higher-order calibration. We have thus shown that machine learned calibrations can help improve the accuracy of xTB-sTDA results over a wide variety of datasets, when compared to a TD-DFT (B3LYP) reference. We can now use these models for various applications. Test set (size) C. Applications of xTB-sTDA calibration 1. Direct vs. calibration ML models ML has been used extensively in the past to explore the excited state space of molecules, primarily being used to directly predict excited state properties such as energies, spectra, and dynamics. However, we expect a ML model trained to directly predict TD-DFT results to perform worse than a calibration model where the baseline method does most of the work and the calibrator simply shifts the result in the right direction. This calibration or ∆-ML approach has been used extensively in the past, and has shown superior performance to pure ML models. Calibration is particularly useful for improved out-of-domain predictive performance. Because supervised ML is a data-driven method, it may have poor performance on molecules distinctly different than those in the training set. In contrast, xTB-sTDA is data-agnostic, so it should give reasonable results regardless, and ML should slightly improve results through calibration.
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To prove this for the xTB-ML models generated in this work, we consider Class 1 ML models trained on the 20k and 300k datasets presented previously, but instead of being trained on the error between TD-DFT and xTB-sTDA (as xTB-ML-(20k,300k) are), the new models, called TDDFT-ML-20k,300k, are trained directly on TD-DFT values. (Note that direct Class 2 ML models would give equivalent results to calibration Class 2 models, since the sTDA energy is provided as input.) xTB-ML-(20k,300k) and TDDFT-ML-20k,300k are then tested on the MOPSSAM 143 dataset, with TDDFT-ML directly predicting values, and xTB-ML predicting the errors which are added to the sTDA energies to get the final cali-
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Predicted S 1 (eV) Predicted S 1 (eV) As seen, the performance of the directly trained ML model is worse than the ML-calibrated xTB data for both dataset sizes. The TDDFT-ML-20k model performs similarly to the linear calibration model (seen in Figure ), while the xTB-ML-20k model already significantly outperforms both. However, it is well known that direct ML models often require more training data than calibration ML models. When expanding the training set to 300k, the TDDFT-ML-300k model outperforms linear calibration but still underperforms compared to both xTB-ML-20k and xTB-ML-300k. Thus, calibrating xTB with ML gives much higher accuracy than using ML to directly predict energies. The benefit to a direct ML model is computational speed, as it can screen approximately 2 orders of magnitude more molecules in a given time period than xTB-ML. However, our goal is to attain approximately the same accuracy as TD-DFT methods, so a direct ML model would not be useful. From the above analysis, our assumption of the improved performance of a calibration ML model is upheld. We now apply our generated calibration ML models for high-throughput screening and chemical space mapping.
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As discussed in the Introduction, one of the motivations of developing this ML calibration is fast and accurate highthroughput virtual screening (HTVS) of spectral conversion materials. The two spectral conversion techniques of interest are triplet-triplet annihilation (TTA) up-conversion and singlet fission (SF) down-conversion. In general terms, TTA involves two sensitizer molecules that absorb low-energy light and transfer their energy to a single emitter molecule which then re-emits the high-energy light. SF involves two emitter molecules, where one absorbs high-energy light and transfers half its energy to a neighboring emitter, and both re-emit lowenergy light. In such molecules, the S 1 excited state is usually involved in absorption/emission while T 1 is typically used for energy transfer. The excited state energy levels of sensitizers and emitters must be well-aligned for efficient spectral conversion -figures of merit (FOMs) to evaluate this alignment are:
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For sensitizers, the first check is if the energies are invalid, i.e. if the T 1 is greater than the S 1 . Then, molecules with S 1 close to T 1 are rated higher. Emitters can be separated into those suitable for singlet fission (singlet more than twice triplet) or triplet-triplet annihilation (singlet less than twice triplet). The same FOM formula is used for both cases, where S 1 close to twice T 1 is desirable. By ensuring the ratios are as close as possible to ideal, we ensure there is minimal loss in energy. Note that properties related to absorption-toemission likelihood such as oscillator strength, triplet-triplet energy transfer probability, triplet-triplet annihilation probability, and others are also important, but are not considered in the present analysis which focuses on optimizing excited state energy level alignment. Although not considered here, xTB-sTDA does output the oscillator strength of each transition, which can be directly used, demonstrating a further benefit of the calibration method.
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We screen the 250k molecules considered in Wilbraham et al. for sensitizers and emitters, to demonstrate the applicability of xTB-ML to high-throughput screening. We use the Class 2 xTB-ML-300k model as it is the most accurate for the MOPSSAM dataset. First, we calibrate S 1 and T 1 energies using the ML model, and compare the results to the linear calibration done in the original work. The results are shown in Figure ).
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For S 1 , the linear calibration is minimal. The ML calibration remains centered around the raw data for mid-to highenergies, but changes more drastically at low energies. This mirrors the previous discussion of Figure , where the linear calibration either over-or under-corrects, but the ML model is more flexible. For T 1 calibration, at mid-to high-energies, both linear and ML calibration shift the energy down, reflecting the tendency of xTB-sTDA to consistently overestimate T 1 energies. For low T 1 energies, the ML model increases the raw energy, suggesting sTDA tends to sometimes spuriously calculate low T 1 energies which can be corrected with ML. Note that because TD-DFT data was not calculated for these 250k molecules, we cannot compare the calibration to ground truth, but based on the metrics presented in Figure , it is likely the ML-calibrated values are more accurate. Now that we have both S 1 and T 1 energies calculated for 250k molecules with xTB-ML, we can identify potential sensitizers and emitters, using the FOMs defined in Equations 1 and 2. Figure ) shows the results of screening molecules for potential sensitizers and emitters.
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As seen, there are several molecules that would function as potential sensitizers and emitters for photon conversion. SI Section VII contains further details about the chemical com-position of the candidate molecules and their distribution in chemical space. The suggested molecules could then be verified with higher-accuracy techniques such as range-separated hybrid TD-DFT or CC2 to confirm their suitability.
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Note here the importance of accuracy for a first-pass screening methodology such as xTB-sTDA. If the uncalibrated results were used, likely several suggested molecules would not be suitable (false positives), and several suitable molecules would not be suggested (false negatives). Using xTB-ML improves the quality of suggestions by reducing both of these rates.
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We have therefore used the Class 2 xTB-ML-300k model to make quick, relatively accurate calculations for S 1 and T 1 energies, and have used the results to screen for potential sensitizers and emitters. This screening was relatively fast as the dataset size was small (250k) and xTB-sTDA results were already provided by Wilbraham et al. For larger datasets on the order of millions (PubChem) or billions of molecules (GDB-17) , running xTB-ML becomes expensive. A more intelligent sampling technique (such as active learning) could be used to screen such large databases, and this is an avenue of future work.
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Since our ML model predicts the error in xTB-sTDA, an interesting application is to map the error in S 1 and T 1 calculations in a global chemical space, to see if there are some areas where xTB systematically over-or under-estimates, or areas where xTB is projected to be fairly accurate. For this analysis, we use Class 1 xTB-ML-300k, as it is shown to be accurate in the general chemical domain and does not require xTB-sTDA computations, so large-scale predictions can be made quickly with ML.
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so a negative error implies xTB-sTDA is over-predicting the excited state energy. As seen in Figure (a), there are distinct regions where xTB-sTDA over-predicts S 1 (right side), regions where xTB-sTDA has reasonable accuracy (top left and center), and regions where it under-predicts (bottom left and top). In general, most molecules are within ±0.5eV error.
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In contrast, for the T 1 energy, xTB-sTDA over-predicts for almost all molecules, as seen in Figure . Note that the scale in this plot is shifted from -0.5-0.5 eV (as in S 1 ) to 0--1.0 eV, to make the distribution of errors clearer. Only a few scattered molecules are under-predicted by xTB-sTDA and are colored red, and all other molecules are over-predicted. Similar to S 1 , xTB-sTDA over-predicts T 1 for most molecules on the right side, and gets reasonable accuracy on molecules in the middle and top left. T 1 is also over-predicted on a cluster of molecules on the bottom left and top.
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Next, we used HDBSCAN to cluster the molecules based on proximity, as shown in Figure ). HDBSCAN takes as input the reduced dimension data from UMAP and outputs a number for each datapoint. It is a soft clustering, not creating distinct categories but instead giving molecules a rating between 0 and 1 (or -1 for no cluster, as approximately 1/3 of the molecules were unable to be clustered) and 100 distinct clusters were created manually from these ratings. We used a minimum cluster size of 10 and the leaf cluster selection method. We can see that HDBSCAN effectively clusters molecules in space, with most molecules in close proximity included in the same cluster. Some of the clusters themselves are spread out across space, such as the purple cluster that includes many molecules along the edge of the global space. Note that this is a dataset-agnostic clustering, as the clustering algorithm only sees molecular information and no labelled data. More details about the HDBSCAN algorithm can be found in their paper and website. A natural question is whether each cluster as defined by HDBSCAN has a particular error associated with it. For example, it seems that xTB-sTDA does a relatively good job for the red cluster, but over-predicts energies for molecules in light blue, purple, and orange clusters. In contrast, the dark green and red clusters seem to have low errors. Although HDBSCAN is a soft clustering, we can categorize molecules into 100 distinct clusters based on the number assigned to them, as well as 2 additional clusters (1 each for unclustered molecules and for outliers). Figure quantifies the mean errors for S 1 and T 1 energies for each cluster.
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Subplots (a) and (b) show the mean errors (ME) of S 1 and T 1 , while (c) and (d) show mean absolute errors (MAE). As seen, the red/yellow/green clusters are likely to have low error, while the purple/dark green clusters have high error. While this analysis is generally useful, the mapping and clustering approach requires knowing the location and cluster categorization of a specific molecule in global chemical space. Oftentimes, this is not known, or would require significant computation.
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Instead, it would be beneficial to have some chemical intuition of accuracy based on the molecular structure, to have greater confidence in xTB-sTDA calculations, or to know to use the ML model or consider other computational techniques. To this end, we can identify substructures that are more likely to be present in low-error or high-error molecules.
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where "under" refers to xTB underestimating the energy while "over" refers to overestimating (note the error definition in Equation ). For both T 1 and S 1 error, we define low error as <±0.05 eV. However, for defining high error, for T 1 we shift the bounds down by 0.5 to reflect the distribution of errors, as seen in Figure (c).
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We can conduct substructure analysis on the low and high overestimation categories (ignoring high underestimation due to low fraction of molecules) to know when to trust the xTB-sTDA results, or when to expect exceptionally high errors. We use molZ to analyze which substructures are overrepresented in each category. The results of this substructure analysis are shown in Figure .
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From these plots, a few patterns become evident. Low error molecules are more likely to be aromatic, potentially with sequential attached rings, for both S 1 and T 1 . In contrast, S 1 high overestimation, S 1 high underestimation, and T 1 high underestimation molecules are likely to be not aromatic, with some unconventional molecular structures included in these groups. In particular, the S 1 overestimation group includes 5 and 7 C-ring molecules, and both S 1 and T 1 underestimation include charged N atoms. There are some aromatic substructures in the T 1 overestimation molecules, but they are attached to the bulk structure with a rotatable bond. This overestimation could be a result of the 3D structure generation, since only limited conformer analysis is conducted and potentially the lowest energy conformer was not achieved. To clarify the effect of this versus an inherent inaccuracy in the excited state energy calculation of xTB-sTDA, a more intensive conformer search could be an avenue of future work. Further substructure analysis of each error category, including most common scaffolds and most common fragments based on RDKit, is provided in SI Section VIII.
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Finally, to show the generalizability of the methodology presented here, we choose a different reference technique beyond TD-DFT, namely CC2. CC2 is known to better predict excitation energies than TD-DFT, but its computational cost is often prohibitively expensive. We use the CC2 S 1 values compiled in QM8, randomly sampling 10k molecules as the training set and using the other 11.5k as the test set. xTB-sTDA values were generated using the same methodology as before. For ML model ensemble generation, because of the smaller dataset, we use 20-fold cross-validation with 95%/5% train/validation splits. This helps ensure all of the data is used in training. As a Class 2 model, both SMILES and sTDA energy are given as input. The new model is termed xTB-CC-ML to distinguish it from the previously generated xTB-ML models.
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Figure (a) shows the results of the comparison, with measurements of accuracy for both methods presented in the inlaid box. As seen, adding the ML calibration to xTB-sTDA results vastly improves results, reducing the MAE by 66%. For comparison, Figure (b) shows the results of TD-DFT calculations using PBE0 and CAM-B3LYP on the same test set of molecules. As seen, xTB-CC-ML has higher accuracy than TD-DFT calculations for the 11.5k test set, using either R2 or MAE as the metric.
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Note that this figure also justifies the main calibration methodology presented in this section, of calibrating xTB-sTDA against TD-DFT. While xTB-sTDA was initially parameterized against mostly CC2 calculations, its accuracy is lower than TD-DFT with hybrid functionals such as PBE0, when compared to CC2. Because TD-DFT values are close in accuracy to CC2 values, calibrating xTB-sTDA to TD-DFT is a useful exercise. The functional B3LYP was chosen in this work due to the large amount of excited state data available using this functional, because it is less computationally intensive than range-separated hybrid functionals such as CAM-B3LYP. Calibrating against more accurate functionals or CC2 could be an avenue of future work.
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To test the impact of training size on accuracy, 8 different ML models were generated with training sizes ranging from 100 to 15,000. The models were then predicted on the remaining molecules in QM8 not used in the training set. The MAE of the test set (against CC2 values) was compared to the MAE of PBE0/def2-TZVP and CAM-B3LYP/def2-TZVP, as shown in Figure ). As seen, a training size of less than 500 molecules allows xTB-sTDA to achieve similar accuracy to PBE0. It is more difficult to match CAM-B3LYP, but this is achieved at a training size of around 1500. At the largest training size considered (15k), xTB-CC-ML vastly outperforms both TD-DFT techniques, with a 62% lower MAE compared to PBE0 and 47% lower MAE compared to CAM-B3LYP. These are promising results; however, the xTB-CC-ML model may not be as generalizable as xTB-ML, due to the smaller (10k), less diverse (only small molecules with up to 8 heavy atoms) training set. To further explore generalizability, two additional ML approaches were considered. First we calibrate xTB-sTDA against CC2 with transfer learning. The learning rate analysis above showed a 10k training set size gives high accuracy while leaving enough molecules in the test set for a reasonable error measurement. Therefore, we train a ML model on 10k randomly sampled molecules from QM8 to predict CC2 values given SMILES and xTB-sTDA energy as input, using the Class 2 xTB-ML-300k model as a starting point, with the first MPNN and first FFNN layers frozen. We use the largest, most detailed ML model considered in this work as a starting point so any adjustments made to this model using the smaller QM8 set should propagate to the larger model. The results of this analysis on the 11k test molecules are shown in Figure . We call this type of model xTB-CC-TL. Our second approach was to calibrate xTB-sTDA against B3LYP, then calibrate B3LYP against CC2. We first generate an ML model that calibrates B3LYP against CC2. We ran B3LYP independently on QM8 using the same settings as outlined in the work. Using these values, we train an ML model on 10k molecules in QM8 that takes in SMILES and B3LYP energy as input and predicts CC2 energy, called B3LYP-CC2-ML. We then apply the Class 2 xTB-ML-300k model generated in this work to predict B3LYP energies. We finally use these predictions as an input to the B3LYP-CC2-ML model to get CC2 energies. We therefore calibrate xTB-sTDA to B3LYP first, and then to CC2. This overall approach is called xTB-B3LYP-CC-ML. The results of this calibration are shown in Figure .
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As seen, both xTB-CC-TL and xTB-B3LYP-CC-ML have similar performance on the test set, and perform slightly worse than xTB-CC-ML. It is difficult to tell a priori which of these models would generalize better, although both would certainly generalize better than the simple xTB-CC-ML model, which does not consider external data at all. For ease of comparison in the future, we have applied all 3 models to the existing datasets. While it would be ideal to have CC2 energies for these datasets, unfortunately this would be prohibitively expensive to generate for the large number of molecules required to obtain a meaningful error value. We have therefore left this analysis for a future work, but have uploaded the ML CC2-calibrated values of all datasets to Github. Overall, xTB-CC-ML serves as a interesting proof of concept that can be expanded further in the future, perhaps with additional CC2 calculations on more diverse molecules.
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We have presented a methodology for calibrating a highthroughput computational chemistry technique (xTB-sTDA) against a high-accuracy one (TD-DFT) using machine learning. We first decided on Chemprop's directed message passing neural network (MPNN) as the ML architecture of choice, then generated a training set using literature scraping of relevant molecules from abstracts (SCOP-PCQC) and an existing excited state database (QM-symex-10k). We also generated an expanded training set using active learning. We built two models based on these training sets (xTB-ML-20k and xTB-ML-300k).
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